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You talk about metric spaces, I assume? Otherwise I wouldn't know what completeness and boundedness mean. Moreover, by boundedness, do you just mean that the metric is bounded? Discrete spaces are locally compact and carry complete bounded metrics without being compact.
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Stefan GeschkeAug 18 '11 at 18:44

By bounded, do you mean finite diameter, or totally bounded? This affects whether or not your question is true.
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Richard RastAug 18 '11 at 18:45

2 Answers
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I don't believe this is true. For example, take an infinite set and put discrete metric on it, that is, $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ if $x\not=y$. Then I believe this is locally compact, complete and bounded, but it is not compact. Did I understand your question correctly?